Right-angled triangle

The right-angled triangle XYZ is similar to the triangle ABC, which has a right angle at the vertex X. The following applies: side a = 9 cm, x=4 cm, x = v-4 (v = height of triangle ABC). Calculate the unknown side lengths of both triangles.

Correct answer:

b =  8.0415 cm
c =  4.0415 cm

Step-by-step explanation:

$$\begin{gathered} a=9\text{cm} \\ x=4\text{cm} \\ k=a/x=9/4=\frac{9}{4}=2.25 \\ {a}^2=b^2+c^2 \\ {x}^2=y^2+z^2 \\ b=c+4 \\ {a}^2=b^2+(b-4{)}^2 \\ {9}^2=b^2+(b-4{)}^2 \\ -2b^2+8b+65=0 \\ 2b^2-8b-65=0 \\ p=2;q=-8;r=-65 \\ D=q^2-4pr=8^2-4\cdot 2\cdot (-65)=584 \\ D>0 \\ {b}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{8\pm \sqrt{584}}{4}=\frac{8\pm 2\sqrt{146}}{4} \\ {b}_{1,2}=2\pm 6.041523 \\ {b}_1=8.041522987 \\ {b}_2=-4.041522987 \\ b=b_1=8.0415=8.0415\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$c=\sqrt{a^2-b^2}=\sqrt{9^2-8.0415^2}=4.0415\text{cm}$

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